The ensemble Kalman filter (EnKF) is a popular data assimilation method in soil hydrology. In this context, it is used to estimate states and parameters simultaneously. Due to unrepresented model errors and a limited ensemble size, state and parameter uncertainties can become too small during assimilation. Inflation methods are capable of increasing state uncertainties, but typically struggle with soil hydrologic applications. We propose a multiplicative inflation method specifically designed for the needs in soil hydrology. It employs a Kalman filter within the EnKF to estimate inflation factors based on the difference between measurements and mean forecast state within the EnKF. We demonstrate its capabilities on a small soil hydrologic test case. The method is capable of adjusting inflation factors to spatiotemporally varying model errors. It successfully transfers the inflation to parameters in the augmented state, which leads to an improved estimation.

Data assimilation combines information from models and measurements into an
optimal estimate of a geophysical field of interest

The ensemble Kalman filter (EnKF)

In hydrology, the EnKF was used for soil moisture estimation from satellite
data

The joint estimation of states and parameters in data assimilation might be
one possibility to reduce the influence of model errors on parameter
estimation

Due to unrepresented model errors and due to a limited ensemble size, the
EnKF underestimates model errors, which can lead to filter inbreeding.
Systematic model errors are common for example in land surface models

Covariance inflation can counteract filter inbreeding. Different methods have
been proposed: (i) additive inflation, which adds a model error after the
forward propagation. This method is especially useful if prior knowledge
about the model error exists. In atmospheric sciences additive inflation has
been successfully applied by, e.g., using reanalysis of historical weather
prediction errors

All these inflation methods are developed in an atmospheric sciences context.
Their transfer to soil hydrology is limited, due to the spatiotemporally
varying model errors and the typically employed augmented state. For
groundwater research,

Alternatively, no inflation method is reported

In this paper, we propose a novel multiplicative inflation method,
specifically designed for the needs in the soil hydrology community. The
inflation method can vary rapidly in space and time to cope with the
typically varying model errors and it is capable of a transfer of the
inflation in the state to the parameters in the augmented state. The
remainder of this paper is organized as follows: Sect.

The EnKF

The forecast propagates an ensemble of states

The state can be extended by, e.g., model parameters

Assuming unbiased Gaussian distributions, the ensemble of augmented states is
characterized through the forecast error covariance matrix

The analysis combines model and measurement information based on the Gaussian
error assumption. The measurement error covariance matrix

The Kalman gain

Based on the measurements, the Kalman gain updates the forecast ensemble to
the analysis ensemble:

This update to the ensemble

Through spurious correlations and non-Gaussian distributions,

A common way to alleviate this issue in hydrology is the use of a damping
factor

Multiplicative inflation is another heuristic way to avoid filter inbreeding.

We propose a more conservative inflation method and ask this question:

In this Kalman filter, the inflation vector is treated as the state variable.
As for parameters, we choose a constant model for the forecast in time:

For the analysis, the distance

The measurement error covariance matrix

The expected distance between measurement and mean forecast based on the
current inflation is

With this Kalman filter, the inflation vector is updated at each time step
based on the difference of the mean forecast to the measurements. Following

We test the proposed inflation method on a small hydrologic test case. We
constructed it specifically to require a strong inflation. This makes it
possible to explore features of the inflation in detail on a rather short
timescale. Due to a small ensemble size, the results vary depending on the
seed of the random numbers. This however, is related to different performance
of the EnKF itself. In simulations (results are not shown), we found that the
behavior of the inflation remains consistent. We have also tested the
inflation method with real-world data by reanalyzing the application by

The Richards equation describes the change of volumetric soil water
content

We additionally consider small-scale heterogeneity through Miller scaling. It
assumes geometrical similarity. With this the microscopic geometry of the
pore space at a macroscopic position is parameterized by a single length
scale

For the test, we choose a one-dimensional case with a depth of 50 cm for a
time of 6 days. We set a groundwater table as the lower boundary condition
throughout the whole time and start from equilibrium conditions. The upper
boundary condition is no flux, except for a rain event with

As material we choose sandy loam from

The forward simulations are performed using MuPhi

To test the inflation method, we perform a perfect model experiment. With the
EnKF we estimate the water content state and four parameters (

Through the unrepresented heterogeneity, we can mimic a model error, leading
to a bias towards smaller values for the estimation of

The EnKF is set up with a total of 25 ensemble members and a damping vector
of

We estimate the water content state together with the four parameters

Water content estimation at the two measurement locations. The standard deviation of the inflated ensemble should be able to explain the differences between the inflated mean and the synthetic truth. The inflation factor is increased when the ensemble uncertainty is too small.

During the first 3 days without any dynamics, the uncertainty for the upper measurement is slightly underestimated, while the uncertainty in the lower measurement is slightly overestimated. This leads to an inflation factor of basically 1 for the lower measurement (factors smaller than 1 are not allowed), while the inflation factor for the upper measurement is larger. However, due to correlations between the measurement locations a stronger inflation to fully explain the difference to the truth is prevented.

The deviation from the synthetic truth is induced through the initial guess of no heterogeneity and can also be seen in the systematic deviation of the inflated mean (which is equal to the forecast mean) from the analysis mean. When the infiltration front reaches the measurements, the deviations from the truth, underestimation of the uncertainty, and inflation factors increase rapidly. All of them are more pronounced for the upper measurement location. After the main peak, the differences and also the inflation factors decrease rapidly again.

The inflation factor for the state is shown in
Fig.

The development of the Miller scaling factors

Inflation factor for the water content state. The inflation is strongest at the upper measurement location during the infiltration, when the uncertainty is underestimated the most. The inflation factor is transferred to the other measurement locations through the correlations in the Kalman gain. The used interpolation of Miller scaling factors impacts these correlations and leads to the smaller inflation directly below the measurement locations.

The initial guess for the scaling factor for the depth of 19.5 cm underestimates the scaling factor, which corresponds to a too fine material. Again, the correlations are small. The value increases slowly during the dry period in the beginning, but is inflated and adjusted strongly during the rain event.

The saturated hydraulic conductivity

The tortuosity

Development of Miller scaling factors

To emphasize the need for a fast-adapting inflation factor, we reduce the
uncertainty of the inflation factors to

The results for the parameters

Development of the inflation factor for the water content state and
saturated hydraulic conductivity

Development of saturated hydraulic conductivity

The proposed inflation method uses a Kalman filter to estimate inflation factors within the EnKF. It is based on the difference between measurements and mean forecast state. It transfers correlations from the forecast of the augmented state to the inflation. Consequently, the performance will be limited if model errors are structurally not represented in the forecast error covariance matrix. The estimation of the inflation factors with a Kalman filter is, like the EnKF itself, based on a linearized analysis. The use of a damping factor can alleviate issues with estimating nonlinear dependent parameters. To keep the inflation consistent with the analysis in the EnKF, we apply the same damping factor for both.

We designed a small synthetic hydrologic test case for the inflation. This test case mimics a model error through initially unrepresented heterogeneity. We designed the test case so that a strong temporally varying inflation is necessary, as it can occur with real data. We choose a short time so that the details of the behavior of the method can be explored. The method showed that it is capable of inflating states and parameters. The inflation is adjusted quickly and differentiates between parameters with strong and not so strong correlations. No over-inflation of weakly correlated parameters occurred. In this specific test case the estimation with inflation is far superior to an estimation without inflation.

The fast adjustment speed of the inflation factor is important because of the
fast-changing model errors and correlations with parameters. The adjustment
speed is determined by the uncertainty of the inflation factor. This
uncertainty is set to a constant value and has to be adjusted. For all our
cases a value of

Fast-dropping correlations between measurements and parameters are a limit for the method. An example could be a parameter only acting on an infiltration boundary condition. After the infiltration is over, correlations with this parameter would drop to zero and the inflation factor for this parameter will not be changed any more. If the inflation factor is not equal to 1 at this time, the parameter spread will keep increasing. In such a case, when there is no correlation, the parameter should be excluded from the estimation and consequently also from the inflation.

The method is in principle capable of compensating unrepresented model
errors. However, it relies on correlations calculated from the forecast
ensemble of the augmented state. If parameters have correlations with
measurement locations with underestimated forecast uncertainties, the
inflation will keep increasing the parameter spread until the forecast
uncertainties are increased sufficiently. Therefore the correlations have to
contain useful information. This means that inflating the parameters based on
their correlations with measurement locations has to increase the forecast
spread at these measurement locations. If the parameters have an insufficient
influence on the state uncertainty, an over-inflation of the parameters can
occur. An example are measurements with underestimated measurement
uncertainties and short times between measurements compared to the timescale
of the dynamics. Then the parameters are not able to increase the state
uncertainty in the short forecast time between measurements and the forecast
dynamics is not able represent the measurement noise. If such errors occur
intermittently, e.g., the closed-eye period as proposed by

In this work we propose a novel spatiotemporally adaptive inflation method, specifically designed for soil hydrology, which nevertheless is expected to work in similar systems as well. The inflation method is based on a Kalman filter acting within the EnKF. The method is capable of rapid adjustments of inflation factors, treating each augmented state dimension individually. This rapid adjustment is required due to temporally varying model errors, as they can appear through violation of the local equilibrium assumption of the Richards equation, hysteresis, or unrepresented heterogeneity.

We demonstrate the use of our inflation method in combination with a damping
factor on a small hydrologic example. We choose heterogeneity as a possible
model error, but allow the heterogeneity to be estimated along with the soil
hydrologic parameters

The synthetic data are available upon request from the corresponding author.

We briefly show the derivation of the Jacobian matrix

We also applied the inflation method to reanalyze the case presented earlier
by

In this paper, we only show the inflation related to the closed-eye period
(Fig.

Figure

Inflation factor for the water content state in a real-world
application for a standard and closed-eye EnKF together with atmospheric
forcing and the corresponding response of measured water contents. The
closed-eye period starts when the infiltration front reaches the topmost TDR.
During this time the local equilibrium assumption of the Richards equation is
violated and a strong inflation is required. The new inflation method allows
a fast adjustment of the inflation factors, which enables the EnKF to
effectively guide the water content states with the TDR measurements through
the closed-eye period. Modified from

HHB designed, implemented, performed and analyzed the presented study. DB provided computational software. All authors participated in continuous discussions. HHB prepared the manuscript with contributions from all authors.

The authors declare that they have no conflict of interest.

We thank editor Insa Neuweiler and two anonymous reviewers for their comments, which helped to improve this paper.

This research is funded by the Deutsche Forschungsgemeinschaft (DFG) through project RO 1080/12-1 and the Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg (Az 33-7533.-30-20/6/2). HGS MathComp provided travel expenses for Hannes H. Bauser and Daniel Berg. Edited by: Insa Neuweiler Reviewed by: two anonymous referees